Doeblin Coefficients and Related Measures
Anuran Makur, Japneet Singh

TL;DR
This paper explores the properties of Doeblin coefficients, revealing their structural, geometric, and contraction characteristics, and introduces new related measures for analyzing multiple distributions and their applications.
Contribution
It presents new structural and geometric properties of Doeblin coefficients, generalizes them to multi-way divergences, and introduces three novel related quantities with applications.
Findings
Doeblin coefficients form a multi-way divergence.
They exhibit tensorization and extremal trace characterization.
New measures like max-Doeblin and DeGroot distances are introduced.
Abstract
Doeblin coefficients are a classical tool for analyzing the ergodicity and exponential convergence rates of Markov chains. Propelled by recent works on contraction coefficients of strong data processing inequalities, we investigate whether Doeblin coefficients also exhibit some of the notable properties of canonical contraction coefficients. In this paper, we present several new structural and geometric properties of Doeblin coefficients. Specifically, we show that Doeblin coefficients form a multi-way divergence, exhibit tensorization, and possess an extremal trace characterization. We then show that they also have extremal coupling and simultaneously maximal coupling characterizations. By leveraging these characterizations, we demonstrate that Doeblin coefficients act as a nice generalization of the well-known total variation (TV) distance to a multi-way divergence, enabling us to…
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Taxonomy
TopicsBayesian Modeling and Causal Inference · Markov Chains and Monte Carlo Methods · Gaussian Processes and Bayesian Inference
